# CHAPTER 6 Quadratic Equations and Inequalities

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```					C H A P T E R 6
Quadratic Equations and Inequalities
Section 6.1                                      Factoring and Extracting Square Roots
Solutions to Odd-Numbered Exercises

1. x2          12x                 35           0                          3.           x2           x           72         0                             5.                       x2           4x           45
x         5 x                 7            0                               x            9 x                  8         0                                      x2          4x               45           0
x         5               x            7                                   x                9                x        8                                   x           9 x                  5            0
x                   9            x         5

7. x2          12x                 36           0                          9. 9x2                    24x              16        0                        11. 4x2               12x               0
x         6 x                 6            0                               3x               4 3x                  4        0                              4x x                    3         0
x         6        0                   x       6       0                   3x               4           0                 3x       4        0             4x              0            x            3           0
x        6                           x       6                                3x                   4                     3x           4             x           0                         x           3
4                                  4
x                3                      x           3

13. u u             9               12 u             9           0         15. 3x x                   6               5x         6        0               17.               y           4 y               3           6
u           9 u              12              0                                x           6 3x               5        0                     y2          7y               12            6          0
u         9           0                u       12          0               x            6           0             3x           5        0                                 y2           7y            6          0
5
u           9                            u       12                           x           6                          x        3                             y           6 y               1           0
y           6            0            y           1       0
y            6                        y       1

19.                 2x 3x                    2         5         6x2       21. x2             64                                                          23. 6x2                   54
6x2               4x         5       6x2                 x            ±           64                                                        x2          9
12x2              4x              5       0                           x            ±8                                                                     x          ±            9
6x            5 2x                     1       0                                                                                                               x          ±3

6x        5            0                   2x          1         0
5                                  1
x               6                        x         2

25. 25x2                16                                                 27. 1 y 2
2                  32                                                      29. 4x 2                  25           0
2           16                                                           2                                                                                                  2
x            25                                                           y            64                                                                            4x               25
16                                                                                                                                                           25
x            ±           25                                               y            ±           64                                                                    x2            4
4                                                                                                                                                                                 25
x            ±5                                                           y            ±8                                                                                    x        ±          4
5
x        ±2

2                                                                             2
31. 4u2             225                  0                                 33. x              4                   169                                     35. x                 3            0.25
225
u2                4                                              x            4           ±     169                                             x           3        ±            0.25
225
u               ±        4                                                   x                4 ± 13                                                       x        3 ± 0.5
15
u               ±2                                                           x           9,     17                                                         x        3.5, 2.5
307
308               Chapter 6                            Quadratic Equations and Inequalities

37. x                     2                                                                      2                                                               2
2            7                                      39. 2x             1            50                               41. 4x         3                98          0
x           2        ±           7                                    2x           1        ±           50                                     4x          3     2
98
x        2±              7                                         2x                   1±5 2                                          4x          3         ±       98
1±5 2                                                    4x          3±7 2
x
2
3±7 2
x
4

43. z2                    36                                              45. x2         4           0                                     47. 9u2           17          0
z               ±            36                                                   x2                4                                               9u2                  17
z               ± 6i                                                               x           ±             4                                                                       17
u          ±
9
x           ± 2i
17
±i
3

2                                                                      2                                                               2
49. t                 3                25                                 51. 3z         4                144              0               53. 2x         3                    54
t           3        ±               25                                            3z           4   2
144                  2x        3           ±            54
t        3 ± 5i                                                            3z           4        ±         144                     2x                3 ± 3i 6
3z           4        ± 12i                                               3 3i 6
x               ±
2   2
3z               4 ± 12i
4 ± 12i
z
3
4
z              ± 4i
3

55. 9 x                       2                                                              2                                                               2
6                121                            57. x          1                    27                           59. x         1               0.04          0

2
121                                      x        1           ±             27                                        x         1   2
0.04
x               6
9
x           1 ± 3i 3                                                    x         1       ±            0.04
121
x           6        ±                                                                                                                                       x            1 ± 0.2i
9
11
x            6±             i
3

2                                                                  2
2                1                                                 7                        38
61.       c                                        0                      63.   x                                                          65. 2x2           5x            0
3                9                                                 3                        9
2                                                                                                       x 2x          5             0
2                   1                                         7                          38
c                                                             x                     ±
3                   9                                         3                          9                  x      0               2x           5       0
5
2                         1                                            7   i                                                                x
c                    ±                                             x              ±              38                                                          2
3                         9                                            3 3
2 1
c            ± i
3 3

67. 2x2                       5x       12              0                  69. x2         900              0                                71. x2        900             0
2x                  3 x              4           0                                     x2           900                                                x2                900
3                                                                              x        ± 30                                                x          ±             900
x                            x               4
2
x          ± 30i
Section 6.1   Factoring and Extracting Square Roots          309

2 2                                                                                              2
73.             x             6                                                                   75. x    5                 100           0
3
2
x           5           100
3       2 2                      3
x             6
2       3                        2                                                                               x           5       ± 10

x2             9                                                                                                    x       15,        5
x           ±3

2                                                                                2
77.           x           5            100            0                                           79. x    2                 18        0
let u          x               5                                                        2
x           2               18
u2           100            0                                                            x           2       ±          18
u           10i u                10i            0                                                                        x           2 ± 3i 2
u       10i             0                           u           10i       0
u                  10i                                  u    10i
x        5                  10i                         x        5    10i
x       5            10i                             x    5      10i

81. Keystrokes:                                                                                                        10

Y           X,T,            x2             9 GRAPH
−15                                            15
x-intercepts are                       3 and 3.
0       x2              9
−10
x           3 x            3
x       3           0             x        3            0
x           3                       x               3

83. Keystrokes:                                                                                                        6

Y           X,T,            x2             2 X,T,                    15 GRAPH               −18                                            18

x-intercepts are                       3 and 5.
0       x2              2x           15
−18
0           x           5 x            3
x       5           0             x        3            0
x           5                       x               3

85. Keystrokes:                                                                                                    6

Y       4                       X,T,                3               x2   GRAPH
−12                                            18

x-intercepts are 1 and 5.
0           4                       2
x         3
2                                                                                        −14
x           3               4
x           3           ±2

x           5, 1
310        Chapter 6                                 Quadratic Equations and Inequalities

5
87. Keystrokes:
Y        2 X,T,                 x2                     X,T,         6 GRAPH
−9                    9
3
x-intercepts are                           2   and 2.
0        2x2           x             6
−7
0         2x           3 x                     2
3
x             2            x             2

6
89. Keystrokes:
−6                    10
Y        3 X,T,                 x2                 8 X,T,            16 GRAPH
4
x-intercepts are                           3   and 4.
2
0        3x            8x                16
−24
0         3x           4 x                     14
4
x            3             x             4

16
91. Keystrokes:
Y         X,T,             x2                  7 GRAPH
0         x2              7
−12
7         x2                                                                                   12

−2
±            7         x
±i       7         x
The equation has complex roots.

7
93. Keystrokes:
Y                 X,T,                     1               x2       1 GRAPH
0            x            1     2             1
2                                                −4
1            x            1                                                                            8

−1
±i            x        1
1±i               x
The equation has complex roots.
18

95. Keystrokes:
Y                 X,T,                     3               x2       5 GRAPH
0             x             3       2        5
−18                   12
5             x             3       2
−2

±             5            x              3
±        5i            x              3
3 ±           5i            x
The equation has complex roots.
2

97. x2          y2           4
y2           4            x2                                                −3                    3

y         ±           4           x2

−2
Section 6.1               Factoring and Extracting Square Roots              311

99. x2                  4y2                4                                                                                                  2

4y2                4            x2
−3                                          3
4            x2
y2
4
4         x2                   4           x2                                    −2
y           ±                                       ±
4                             2

101.                                                  x4              5x2           4           0
x2              4 x2              1            0
x            2 x                 2 x                  1 x              1            0
x               2           0                x            2        0                x           1    0   x   1   0
x           2                             x             2                       x    1       x       1

103.                x4              5x2              6            0                                                               105.        x4        3x2          4        0
x2              3 x2                 2                0                                                                      x2        4 x2          1            0
x2               3           0                            x2        2           0                                                x2        4        0            x2       1           0
x2           3                                      x2          2
x2           4                     x2              1
x           ±           3                             x        ±           2
x        ±2                        x       ±i

107.                x2              4    2
2 x2              4            3           0
x2               4               3           x2           4             1           0
x2              1 x2               5           0
x2               1            0                           x2         5           0
x2               1                                      x2          5
x            ±1                                        x        ±           5

?
109.                            x           7 x                   10        0                                                     Check: 25             7 25                 10           0
?
let u                        x                                                                                                                 25       35           10           0
2
x                7           x                10           0                                                                                              0           0
u2           7u               10           0                                                                                                  ?
Check:           4        7 4              10           0
u           5 u                  2            0                                                                                                  ?
4        14           10           0
u            5                   u            2                                                                                                               0           0
x            5                           x            2
x            25                          x            4

111.                x2       3           x1      3           6         0
x1       3            3 x1            3           2            0
x1   3            3              0                    x1   3         2           0
x1 3                3                              x1 3                 2
x           27                                  x               8
312             Chapter 6                                    Quadratic Equations and Inequalities

113.           2x2      3               7x1      3           5         0                                          115.           x2    5        3x1      5           2           0
let u                x1      3                                                                                   x1    5         2 x1        5           1            0
2    x1 3 2                   7x1 3                   5         0                                                x1 5           2                x1 5            1
2u2               7u                 5         0                                                     x         25                   x           15
2u              5 u                  1             0                                                     x         32                   x           1
2u              5            0                         u            1       0
2u              5                                      u       1
5
u            2                              x1 3            1
5
x1       3
2                             x1   3           13
5 3
x1   3
2                                  x       1
125
x             8

1          3
117.               2x2      5            7x1         5           3         0                                      119.                                      2           0
x2         x
2x1 5                    1        x1 5                3             0
1         3x           2x2             0
1
x1      5
x1      5
3                                                     2x2           3x           1           0
2
5                                                                                  2x            1 x            1              0
1
x                                                x         35
2                                                                                       2x         1        0               x           1           0
1                                                                                                              1
x                                                x         243                                                         x                                    x           1
32                                                                                                              2

121.           x           5) x                          2             0                                          123.       x             1            2           x            1               2           0
x               5 x               2            0                                                     x         1                 2           x              1                2       0
2
2
x2           5x               2x              10            0                                                                                        x           1                    2           0
x2            3x              10            0                                                                                        x2          2x               1       2       0
x2           2x          1       0

125. x                  5i x                             5i                0                127.     0     16t2       256                                           129.                0            16t2         128
x            5i x                 5i                0                       16t2   256                                                                    16t2            128
x2            25i2                 0                         t2   16                                                                            t2       8
x2           25               0                          t   4 seconds                                                                      t       ±       8
t       ±2       2
t       2 2             2.828 seconds

131. 0                  144                 128                  162                                              133. 1685.40                      1500 1                  r    2

0                16t2                  128t                   144                                                     1.1236                 1           r    2

0                16 t2                    8t              9                                                                   1.06           1           r
0                16 t                  9 t                1                                                                    .06           r
t           9            0                            t        1            0                                                  6%            r
t            9 seconds                              t               1
Section 6.2                         Completing the Square                           313

135.                     892                  26.6           t   2

892              26.6           t
892           26.6              t
3           t
Year 1993

137. (a) h0             100 feet                    v0           0 feet sec              h    0   (b) h0            100 feet                   v0        32 feet sec                          h   100 feet
0        16t 2                0      t           100                                  100                 16t2             32t           100
16t2          100                                                                                0             16t2             32t
t2       6.25                                                                               0             16t t                2
t             6.25                                                                       16t           0                              t        2           0
t     2.5 seconds                                                                            t         0 seconds                               t           2 seconds
Extracting the roots method was used because the                                           Factoring method was used because the quadratic equation
quadratic equation did not have a linear term.                                             did not have a constant term.

139. Factoring and the Zero-Factor Property allow you to                                          143. To solve an equation of quadratic form, determine an
solve a quadratic equation by converting it into two                                              algebraic expression u such that substitution yields the
linear equations that you already know how to solve.                                              quadratic equation au2 bu c 0. Solve this
quadratic equation for u and then, through back-substitu-
tion, find the solution of the original equation.

141. False. The solutions are x                                      5 and x             5.

Section 6.2                               Completing the Square

8       2                                                                                                    20      2
1. x2        8x        16                    16                                                   3. y2            20y            100                 100
2                                                                                                            2

16      2                                                25                   25            5    2
5. x2        16x            64                64                                                  7. t 2           5t
2                                                        4                     4            2

81                   81                 9       2                                          1              1                 1                    1       1   2
9. x2        9x                                                                                  11. a2              a
4                     4                 2                                                  3             36                36                    3       2

3           9                         9                        3   1   2                                                                                            0.4     2
13. y 2         y                                                                                 15. r2            0.4r           0.04                  0.04
5          100                       100                       5   2                                                                                                 2

17. (a) x2             20x           100            100                                           19. (a) x2               6x          9           0         9
x           10   2         100                                                                   x        3   2           9
x        10            ± 10                                                                      x        3           ±3

x         10 ± 10                                                                            x                3±3
x         20, 0                                                                              x                6, 0
(b) x2           20x          0                                                                (b) x2                  6x           0
xx            20           0                                                                          xx            6            0
x         0            x            20                                                                x         0            x           6         0
x             6, 0

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